Blow-up profiles for the parabolic-elliptic Keller-Segel system in dimensions $n\ge 3$

TL;DR

This paper analyzes the asymptotic behavior of radially decreasing solutions of the Keller-Segel system in dimensions three and higher, revealing a universal blow-up profile characterized by a specific power-law decay.

Contribution

It establishes the asymptotic profile of blow-up solutions in higher dimensions and contrasts it with the two-dimensional case, extending understanding of blow-up phenomena.

Findings

Final blow-up profile behaves like |x|^{-2} in higher dimensions.

Refined estimates for type I blowup solutions near blow-up time.

Existence of self-similar blowup solutions in dimensions n≥3.

Abstract

We study the blow-up asymptotics of radially decreasing solutions of the parabolic-elliptic Keller-Segel-Patlak system in space dimensions $n\ge 3$. In view of the biological background of this system and of its mass conservation property, blowup is usually interpreted as a phenomenon of concentration or aggregation of the bacterial population. Understanding the asymptotic behavior of solutions at the blowup time is thus meaningful for the interpretation of the model. Under mild assumptions on the initial data, for $n\ge 3$, we show that the final profile satisfies $C_1|x|^{-2}\le u(x,T)\le C_2|x|^{-2}$, with convergence in $L^1$ as $t\to T$. This is in sharp contrast with the two-dimensional case, where solutions are known to concentrate to a Dirac mass at the origin (plus an integrable part). We also obtain refined space-time estimates of the form $u(x,t)\le C(T-t+|x|^2)^{-1}$ for type~I blowup solutions. Previous work had shown that radial, self-similar blowup solutions (which satisfy the above estimates) exist in dimensions $n\ge 3$ and do not exist in dimension $2$. Our results thus reveal that the final profile displayed by these special solutions actually corresponds to a much more general phenomenon.